Variational Autoencoder task for better feature extraction
I have not worked on this but I think I can give you a theoretical perspective of using VAE's. Regression is a Supervised Learning task and is basically a mapping from Input to Output where the Neural Net will approximate the function $f(input) = output$.

VAE's on the other hand are good for finding how a latent variable affects the output. For example, if you have a task of training on a persons facial emotions, and if your latent space contains 2 variables $z_1$ and $z_2$ then you might find varying $z_1$ varies the amount of smile on the face, while varying $z_2$ might give the amount of drooping of eyes. I suggest you check this video from Stanford at ~44:00 to see this actually happens or check this blog. So VAE's might have been useful if your output contained more features which would vary according to variations in latent variables, but a single scalar output can only tell you about the rate of effect on varying a latent variable.

But if your job is only for better regression, Auto-Encoders are the better alternative, since it has an inherent de-noising ability and sufficient training might help in de-noising the input, and thus provide better results if classified on the basis of latent variables. 

An approach, which I think is kind of similar (to your thinking) has been proposed by Kingma, et al. for Semi Supervised learning in this paper. The paper has very poor description of the method so I would suggest you check out this blog. They have used an additional classifier for reconstruction of the original input and trained the classifier when labels are present.